scholarly journals Radial kinetic nonholonomic trajectories are Riemannian geodesics!

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Alexandre Anahory Simoes ◽  
Juan Carlos Marrero ◽  
David Martín de Diego
Keyword(s):  
Fixed Point ◽  
Riemannian Metrics ◽  
Nonholonomic Systems ◽  
Nonholonomic Mechanics ◽  
Exponential Map ◽  
Surprising Result ◽  
Prove Theorem ◽  
Starting Point ◽  
Small Times

AbstractNonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed point q are true geodesics for a family of Riemannian metrics on the image submanifold $${{\mathcal {M}}}^{nh}_q$$ M q nh of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point q, for sufficiently small times, minimize length in $${{\mathcal {M}}}^{nh}_q$$ M q nh !

scholarly journals Fields Connected with Particles in Terms of Complex-Riemannian Geometry

1990 ◽  
Vol 45 (2) ◽  
pp. 81-94
Author(s):  
Julian Ławrynowicz ◽  
Katarzyna Kędzia ◽  
Leszek Wojtczak
Keyword(s):  
Field Potential ◽  
Riemannian Metrics ◽  
Interaction Matrix ◽  
Explicit Calculation ◽  
Maxwell System ◽  
Curved Space ◽  
Starting Point ◽  
Autocorrelation Time ◽  
The Fourier Transform ◽  
Convolution Equations

AbstractA complex analytical method of solving the generalised Dirac-Maxwell system has recently been proposed by two of us for a certain class of complex Riemannian metrics. The Dirac equation without the field potential in such a metric appeared to be equivalent to the Dirac-Maxwell system including the field potentials produced by the currents of a particle in question. The method proposed is connected with applying the Fourier transform with respect to the electric charge treated as a variable, with the consideration of the mass as an eigenvalue, and with solving suitable convolution equations. In the present research an explicit calculation based on linearization of the spinor connections is given. The conditions for the motion are interpreted as a starting point to seek selection rules for curved space-times corresponding to actually existing particles. Then the same method is applied to solids. Namely, by a suitable transformation of the configuration space in terms of elements of the interaction matrix corresponding to the Coulomb, exchange, and dipole integrals, the interaction term in the hamiltonian becomes zero, thus leading to experimentally verificable formulae for the autocorrelation time

scholarly journals Riemannian metrics on Lie groupoids

2018 ◽  
Vol 2018 (735) ◽  
pp. 143-173 ◽  
Author(s):  
Matias del Hoyo ◽  
Rui Loja Fernandes
Keyword(s):  
Riemannian Metrics ◽  
Exponential Map ◽  
Important Class ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Linearization Theorem

AbstractWe introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein–Zung linearization theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.

scholarly journals Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Fixed Point ◽  
Friction Factor ◽  
Fixed Point Method ◽  
Point Method ◽  
Symbolic Form ◽  
Pipe Surface ◽  
Flow Friction ◽  
Relative Roughness ◽  
Starting Point ◽  
Derivatives Of

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems

1961 ◽  
Vol 13 ◽  
pp. 78-82 ◽  
Author(s):  
Smbat Abian ◽  
Arthur B. Brown
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Prove Theorem ◽  
The Fixed Point Theorem ◽  
The Axiom Of Choice

In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries.Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}.Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.

scholarly journals COUNTING FIXED POINTS, TWO-CYCLES, AND COLLISIONS OF THE DISCRETE EXPONENTIAL FUNCTION USING p-ADIC METHODS

2012 ◽  
Vol 92 (2) ◽  
pp. 163-178 ◽  
Author(s):  
JOSHUA HOLDEN ◽  
MARGARET M. ROBINSON
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Exponential Function ◽  
Discrete Logarithm ◽  
Exponential Map ◽  
Hensel’S Lemma

AbstractBrizolis asked for which primes p greater than 3 there exists a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Various authors have contributed to the understanding of this problem. In this paper, we use p-adic methods, primarily Hensel’s lemma and p-adic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.

On the Geometry of Nonholonomic Dynamics

1994 ◽  
Vol 61 (3) ◽  
pp. 689-694 ◽  
Author(s):  
H. Esse´n
Keyword(s):  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Vector Equation ◽  
Finite Degree ◽  
Abstract Form ◽  
Nonholonomic Dynamics ◽  
Starting Point ◽  
Projection Approach ◽  
Local Tangent ◽  
Lagrange’S Method

The formulation and derivation of equations of motion for finite degree-of-freedom nonholonomic systems, is discussed. The starting point is Newton’s equation of motion in the 3K-dimensional unconstrained configuration space of K particles. Constraints represent knowledge that motion is only possible along some directions in the local tangent spaces. Only projections of the 3K-dimensional vector equation onto these allowed directions are of interest. The formalism is essentially that of Kane-Appell cast into an abstract form. It is shown to give the same equations as Hamel’s generalization of Lagrange’s method. The algorithmic advantage of the Kane-Appell projection approach is stressed.

scholarly journals Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Computation ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 48 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Fixed Point ◽  
Relative Error ◽  
Symbolic Regression ◽  
Logarithmic Function ◽  
Flow Friction ◽  
First Order ◽  
Quasi Monte Carlo ◽  
Initial Stage ◽  
Starting Point ◽  
Transcendental Functions

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.

scholarly journals Effective Uniform Bounds on the Krasnoselski-Mann Iteration

2000 ◽  
Vol 7 (9) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Convex Sets ◽  
Fixed Point Theory ◽  
Uniformly Convex ◽  
Asymptotic Regularity ◽  
Normed Spaces ◽  
Mann Iteration ◽  
Starting Point

This paper is a case study in proof mining applied to non-effective proofs<br />in nonlinear functional analysis. More specifically, we are concerned with the<br />fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general so-called Krasnoselski-Mann iterations. These iterations converge to fixed points of f only under special compactness conditions and even for uniformly convex<br />spaces the rate of convergence is in general not computable in f (which is<br />related to the non-uniqueness of fixed points). However, the iterations yield<br />approximate fixed points of arbitrary quality for general normed spaces and<br />bounded C (asymptotic regularity).<br />In this paper we apply general proof theoretic results obtained in previous<br />papers to non-effective proofs of this regularity and extract uniform explicit<br />bounds on the rate of the asymptotic regularity. We start off with the classical<br />case of uniformly convex spaces treated already by Krasnoselski and show<br />how a logically motivated modification allows to obtain an improved bound. Already the analysis of the original proof (from 1955) yields an elementary<br />proof for a result which was obtained only in 1990 with the use of the deep<br />Browder-G¨ohde-Kirk fixed point theorem. The improved bound from the modified<br /> proof gives applied to various special spaces results which previously had<br />been obtained only by ad hoc calculations and which in some case are known<br />to be optimal.<br />The main section of the paper deals with the general case of arbitrary normed<br />spaces and yields new results including a quantitative analysis of a theorem<br />due to Borwein, Reich and Shafrir (1992) on the asymptotic behaviour of<br />the general Krasnoselski-Mann iteration in arbitrary normed spaces even for unbounded sets C. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform<br />bounds which do not depend on the starting point of the iteration and the<br />nonexpansive function and the normed space X and, in fact, only depend<br />on the error epsilon, an upper bound on the diameter of C and some very general information on the sequence of scalars k used in the iteration. Even non-effectively only the existence of bounds satisfying weaker uniformity conditions was known before except for the special situation, where lambda_k := lambda is constant. For the unbounded case, no quantitative information was known so far.

scholarly journals The proof-theoretic analysis of transfinitely iterated quasi least fixed points

2006 ◽  
Vol 71 (3) ◽  
pp. 721-746 ◽  
Author(s):  
Dieter Probst
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Theoretic Analysis ◽  
Starting Point

AbstractThe starting point of this article is an old question asked by Feferman in his paper on Hancock's conjecture [6] about the strength of . This theory is obtained from the well-known theory ID1 by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of also permit to analyze its transfinitely iterated variants . Thus, we eventually know that

On the directly generated resonant standing waves in a rectangular tank

1990 ◽  
Vol 217 ◽  
pp. 143-165 ◽  
Author(s):  
Lev Shemer
Keyword(s):  
Experimental Data ◽  
Fixed Point ◽  
Steady State ◽  
Numerical Study ◽  
Standing Waves ◽  
Stationary Solutions ◽  
Damping Coefficients ◽  
Rectangular Tank ◽  
Starting Point ◽  
The Stability

A numerical study based on the nonlinear Schrödinger equation, as applied to nonlinear resonant standing waves excited directly by a wavemaker in a rectangular tank, is presented. The stationary solutions of the problem serve as a starting point of the investigation. Bifurcations from a single steady state to multiple stationary solutions are obtained for several values of damping coefficients along the tank and at the wavemaker. The stability of the latter solutions is tested. Limit-cycle or fixed-point solutions are obtained. The results of the numerical study are discussed in connection with experimental data. The necessity of incorporation of dissipation at the wavemaker in the theoretical model in order to obtain qualitative agreement with experiment is demonstrated.

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